Le i a ion o supe conduc i e cable in ea h magne ic ield
267
LEVITATION OF SUPERCONDUCTIVE CABLE
IN EARTH MAGNETIC FIELD
P. Ka ban
a)
, I. Doležel
b)
,
B. Ul ych
a)
a)
Facul y o Elec ical Enginee ing UWB, Sady P a icá ník 14, 314 06 Plze, E-mail:{ ul ych, ka ban}@k e.zcu.cz
b)
Ins i u e o The momechanics ASCR, Dolejško a 5, 182 00 P aha 8, E-mail: [email protected]
Summa y The pape ep esen s an in oduc o y s udy abou a supe conduc i e cable le i a ing in Ea h’s magne ic ield.
Buil a e wo ma hema ical models o he p oblem p o iding bo h he shape o he a c o he cable and o ces ac ing along i .
The heo e ical analysis is supplemen ed wi h an illus a i e example.
1. INTRODUCTION
Ea h’s magne ic ield is p oduced by o a ion
o liquid and elec ically conduc i e shell 3 o
Ea h’s co e 2 wi h espec o ela i ely unmo able
shell 1, see Fig. 1a (ano he iew is in Fig. 1b).
Fig. 1a: Ea h magne ic ield
This ield whose s eng h on he no he n
hemisphe e eaches alues be ween 38–56 A/m
(which depends on he s uc u e o he
co esponding li hosphe ic pla es) ep esen s one o
e y impo an a ibu es o he plane Ea h. The
human popula ion employs his ield
• unconsciously – he ield ep esen s an
“umb ella” p o ec ing Ea h’s su ace agains
he impac o elec ically cha ged pa icles
a i ing o Ea h in he consequence o he sola
ac i i y and also om space and
• consciously – Ea h’s magne ic ield is used o
o o ien a ion on Ea h’s su ace – he i s
magne ic compasses we e buil in China,
abou 600 yea s be o e Ch is ,
o geological su ey – subsu ace deposi s o
e omagne ic o es p oduce local anomalies
in Ea h’s magne ic ield,
o de e mina ion o he ime scales in geology
and also in a cheology.
Nowadays, howe e , we can o en see a ious
conside a ions aimed a i s non adi ional
employmen . Men ioned can be, o example,
„ e he s“ (pa icula s can be ound in [1] and [2]),
which a e supe conduc i e cables unwound in a
sui able di ec ion om a ious space objec s
(sa eli es, he las sec ions o boos e ocke s, space
labs e c.) ha can be used as:
• Elec ic ol age sou ce – a sui ably o ien ed
conduc o o leng h
l
mo ing a a eloci y
in
magne ic ield o lux densi y
B
induces
ol age
( )
i0
d
l
u
= × ⋅

B l
[3]. T anspo o
such a cable o he o bi is much cheape han
anspo o classical pho o ol aic cells.
• A sui ably o ien ed conduc o o leng h
l
ca ying cu en
I
and mo ing in magne ic
ield o lux densi y
B
is a ec ed by he
Lo en z o ce o alue
( )
L0
d
l
I= ×

F l B
[3]
ha deccele a es o accele a es he space objec
and shi s i o a lowe o highe o bi .
These non adi ional space applica ions ha a e
in ensi ely s udied and now al eady also
expe imen ally alida ed (see, o example, [1] –
p ojec s TSS-1a Oidipus-C, NASA+I alian Space
Agency) could also be ans e ed o Ea h’s su ace.
The pape deals wi h one possible applica ion –
possibili y o employmen o Ea h’s magne ic ield
o le i a ion o a supe conduc i e cable closely
abo e Ea h’s su ace, which could be used, o
ins ance, in me eo ology, whe e he le i a ing cable
could eplace he ial balloons, pa icula ly o
in es iga ion o lowe laye s o he a mosphe e.
Fig. 1b: Ea h magne ic ield wi h he indica ed posi ion o
he cable (see Figs. 3a, b)
Ad ances in Elec ical and Elec onic Enginee ing
268
2. FORMULATION OF THE PROBLEM
Fig. 2. depic s an a angemen o a ypical su-
pe conduc i e cable o ea hly condi ions. A simila
cable could play a ole in he conside ed case.
Fig. 2: A angemen o he supe conduc i e cable
I s supe conduc i e sys em consis s o a g ea e
amoun o hin Cu ubes 2 illed in wi h he ac ual
supe conduc o and placed in Ke la shell 1. The
shell ep esen ing he ca ying elemen is lown
h ough liquid He 3 ha secu es he supe conduc i e
egime o he cable.
The cable o s a ing leng h
0
2
l
≡
s is loca ed
on Ea h’s su ace (nea he equa o ) in he ho izon-
al posi ion be ween poin s P and Q (see Fig. 1b)
whose abscissa is o ien ed pe pendicula ly o he
o ce lines o Ea h’s magne ic ield. Nea he equa-
o we can conside Ea h’s magne ic ield app oxi-
ma ely pa allel o Ea h’s su ace, so ha i s lux
densi y
0
B
=
z
B (see Fig. 3a).
Fig. 3a: The in es iga ed a angemen o a supe conduc -
ing cable in he domain wi h uni o m
lux densi y – s a ing posi ion
The poin s P and Q ep esen he eel d ums
equipped wi h b akes. The cable is wound on hem
and a e hei b aking o i can eely unwind o a
gene al leng h
2
s l
>
(Fig. 3b). I he cable ca ies
cu en
I
in he indica ed di ec ion, i begins o be
a ec ed by he o al Lo en z o ce
( )
L
d
l
l
I
−
= ×

F l B
o ien ed in di ec ion
y
.
This o ce hen li s he cable upwa ds o a gen-
e al posi ion
(
)
y x
= co esponding o he “b aked
o ” leng h
s
o he cable. In his posi ion, howe e ,
he speci ic Lo en z o ce ac ing on he uni leng h
o he cable has gene ally wo componen s
(
)
L 0 L, 0 L,
x y
= ± +x y
.
Thus, he le i a ion e ec
o Ea h’s magne ic ield can be expec ed o de-
c ease wi h g owing leng h
s
o he cable.











Fig. 3b: The in es iga ed a angemen – inal posi ion o
he cable
The aim o he pape is o e alua e he
• dependence
(
)
y x
= on he “b aked o ”
leng h
s
o he cable and on he alue
I
o he
supe conduc i e cu en ,
• dependence o he dis ibu ion o o ces ac ing
on he cable on he same pa ame e s,
• es ic ion o he conside ed le i a ing e ec by
he “b aked o ” leng h
s
and cu en
I
.
3. MATHEMATICAL MODELS OF THE
PROBLEM AND ITS SOLUTION
The ma hema ical desc ip ion o he p oblem
may be ca ied ou in wo ways:
• Solu ion o a nonlinea o dina y di e en ial
equa ion o mula ed as an ini ial p oblem and
exp essing he o ce and o que balance in an in-
ini esimal elemen o he cable.
• Solu ion o a sys em o algeb aic equa ions
exp essing only he balance o o ces in a ini e
elemen o he cable ( he o que balance is he e
sa is ied au oma ically)
Each o hese wo models has i s ad an ages and
d awbacks. The di e en ial model allows using o
nume ical algo i hms o p o essionally sophis ica ed
unc ion p ocedu es p o iding bo h con e gence and
equi ed accu acy o solu ion o he co esponding
equa ion. On he o he hand, he algeb aic model is
mo e lexible and p o ides an easie ealiza ion o
some pa ial, speci ic compu a ions such as local
dis ibu ion o ex e nal o ces, supp ession o de-
o ma ion o pa s o he supe conduc i e cable e c.
Tha is why we used bo h models ha p o ided a
good acco dance o he esul s.
Le i a ion o supe conduc i e cable in ea h magne ic ield
269
3.1. Di e en ial ma hema ical model
Fo ob aining he di e en ial equa ion we s a
om Fig. 4a and Fig. 4b showing he li ed cable
and si ua ion in i s elemen .
Fig. 4a: The li ed cable wi h an elemen
He e
1 2
,
H H
deno e he ho izon al componen s
o o ces a poin s 1 and 2,
1 2
,
V V
he e ical com-
ponen s and
q
is he uni weigh o he cable.
Fi s le us exp ess he pa icula componen s o
he Lo en z o ces. F om Fig. 4b we can easily de-
i e ha
L L
2 2
1
d d d d
1 1
x y
y'
F BI s , F BI s
y' y'
= − ⋅ = ⋅
+ +
(1)
Fig. 4b: De ailed si ua ion in an elemen be ween poin s 1
and 2 (see Fig. 2a)
He e
1 2
,
H H
deno e he ho izon al componen s
o o ces a poin s 1 and 2,
1 2
,
V V
he e ical com-
ponen s and
q
is he uni weigh o he cable ha
can gene ally be qui e nonuni o m.
Fi s le us exp ess he pa icula componen s o
he Lo en z o ces. F om Fig. 4b we can easily de-
i e ha
L L
2 2
1
d d d d
1 1
x y
y'
F BI s , F BI s
y' y'
= − ⋅ = ⋅
+ +
(1)
and as
'2
d 1 d
s y x
= + we immedia ely ha e
L L
d d d d
x y
F BI y, F BI x
= − =
.
(2)
The balance equa ions o he o ces in he ele-
men ead
1 L 2
d
x
H F H
+ =
,
1 L 2
d d
y
V F q s V
− − =
(3)
and o he o que wi h espec o poin 2
L 1 L 1
d d d
d d d d d
2 2 2
y x
x y x
F V x F q s H y
⋅ − + ⋅ = ⋅ − . (4)
As a any poin o he cable
V Hy'
=
, we im-
media ely ha e
(
)
d d d d
V Hy' H y' y' H
= = ⋅ + (5)
whe e (see (3))
2 1 L
d d
x
H H H F
= − =
,
2 1 L
d d d
y
V V V q s F
= − = − .
A e subs i u ing om (2) and o
d
s
we inally
ha e
2
1 d d d d
'
q y x BI x H y' y' BI y
+ − = ⋅ +
and hence
(
)
2 2
1 1
'
q y BI y' H y''
+ − + = ⋅
, (6)
which ep esen s a nonlinea di e en ial equa ion
desc ibing he esul an shape o he cable. Fo ce
H
is gi en as (see (2))
0
H BIy H
= + (7)
whe e
0
H
is he ho izon al ension in he cable a i s
beginning (
x l
= ±
). The i s ini ial condi ion ead
(
)
(
)
0
y l y l
= − =
, he second one is he known
leng h
s
o he cable.
3.2. Algeb aic ma hema ical model
Conside an a angemen o he supe conduc i e
cable in Ca esian coo dina e sys em as is depic ed
in Fig. 3a. In di ec ion
x
he dis ance
/ 2
l
is di-
ided in o
e
N
uni o m elemen s
x
∆
(Fig. 5a) wi h
an equi alen numbe o co esponding nonuni o m
elemen s
i
s
∆
on he cu e
(
)
s y x
≈ desc ibing he
shape o a c o he cable. To a gene al dis ance
i
x
we assign he gene al leng h
i
s
o he co esponding
a c o he cable.
   











Fig. 5a: Dis ibu ion o o ces on he cable
The pa
i
s
o he cable is a ec ed by he ol-
lowing o ces (Fig. 5b):
Ad ances in Elec ical and Elec onic Enginee ing
270
•
s0
F
,
s,
i
F
– in e nal o ces p oducing ension in
he cable,
•
L,
i
– he Lo en z o ces ac ing pe pendicula ly
on pa icula elemen s
i
s
∆
,
•
g,
i
F
– weigh o pa icula elemen s
i
s
∆
,
•
ex
F
– ex e nal o ce loading he cable.
Fig. 5b: Fo ces in an elemen o he cable
Acco ding o Figs. 5a, 5b we ha e
s0 0 s0 ex 0 ex
s, 0 s , 0 s , s0 s0
ex ex g, 0 g , g , g
s , s, s , s,
, ,
, ,
, , ,
cos , sin .
i x i i
i i i i
x i i i i i i
F F
F F F F
F F F s
F F
ψ
ψ ψ
ψ
α α
= − =
= + =
= = = ∆
= =
x
x
F F
F
F F
F F
ψ
ψψ
ψ
ψ
ψψ
ψ
ψ
ψψ
ψ (8)
A he same ime he e holds
2 2
e
1
, ,
sin , cos .
i i
i
i i
i i
x s x
N
x
s s
ψ
ψ
α α
∆ = ∆ = ∆ + ∆
∆∆
= =
∆ ∆
(9)
The condi ions o balance o pa icula o ces
now ead
0 L , s ,
1
ex L , s , g ,
1 1
: 0,
: 0.
i
s x k x i
k
i i
k i k
k k
x F F
F F
ψ ψ ψ
ψ
=
= =
− + + =
− + + =

 
(10)
A e subs i u ion om (8) and (9) o (10) we ob ain
s0 s,
1
s0
1
s,
sin cos 0
sin
cos
i
k k i i
k
i
k k
k
ii
F BI s F
F BI s
F
α α
α
α
=
=
− + ∆ + = 
− ∆
=


(11)
Analogously o he componen s in di ec ion
ψ
we
ha e
ex g S,
1 1
s, ex g
1 1
cos sin 0
cos / sin .(12)
i i
k i k i i
k k
i i
i k i k i
k k
F BI s s F
F F BI s s
α α
α α
= =
= =
− ∆ + ∆ + = 
 
= − + ∆ − ∆
 
 
 
 
 
Compa ison o (11) and (12) p o ides
s0
1
ex g
1 1
sin
cos
cos
sin
i
k k
k
i
i i
k i k
k k
i
F BI s
F BI s s
α
α
α
α
=
= =
− ∆
=
− + ∆ − ∆

 
(13)
The algo i hm o solu ion o his ma hema ical
model may be ealized in he ollowing s eps:
• Fo some alue o
s0
F
solu ion o (13) by, o
example, he Regula Falsi me hod using (9),
p o ides, by means o [4], he alue o
i
ψ
.
• Solu ion o (11) wi h espec o (8) p o ides o
he alue
i
ψ
he o ce
S,
i
F
.
• The i s wo s eps a e epea ed o all alues o
e
1, ,
i N
=K ; a he same ime we calcula e he
alue o
e
1
N
k
k
s
=
∆

ha is necessa y o co ec ion
o
s0
F
.
• Condi ion
e
1
N
k
k
s s
=
∆ =

hen p o ides (again using
he Regula Falsi me hod) he alue o
s0
F
.
4. COMPUTER MODEL, ACHIEVED AC-
CURACY OF SOLUTION
Solu ion o he ma hema ical model 3.1 (equa-
ion (6)) was ca ied ou by he ou h-o de Runge-
Ku a me hod w i en in Ma lab. P o ed was e y
good con e gence o he nume ical p ocess – o
inding o
(
)
y x
in he in e al
0 50
x l
≤ ≤ =
m
wi h accu acy o 3 alid digi s we needed only 50
s eps.
Solu ion o he ma hema ical model 3.2 was pe -
o med by a use p og am w i en in Bo land Delphi.
The con e gence o he nume ical p ocess o ind-
ing bo h
(
)
y x
and
(
)
S
x
F was good again. Accu-
acy o 3 alid digi s was eached in abou 200 s eps.
5. ILLUSTRATIVE EXAMPLE
5.1. Technical speci ica ion
Conside ed is an a angemen o he supe con-
duc ing le i a ing cable acco ding o Fig. 2. I s di-
mensions and all physical pa ame e s a e lis ed in
Tab. 1. I is necessa y o ca y ou such a se o es -
ing compu a ions ha would p o ide answe s o
ques ions in pa ag aph 2.
5.2. Resul s and hei discussion
Dependencies
(
)
y x
≡s on he b aked-o
leng h o he cable and cu en
I
ollow om Fig. 6.
Table 2: Physical pa ame e s o he cable
quan i y symbol uni alue
Le i a ion o supe conduc i e cable in ea h magne ic ield
271
dis ance PQ
2
l
m 100
leng h o
b aked-o cable
s
m 120
130
140
ex e nal o ce
ex
F
N 0 (*)
speci ic weigh
g
N/m 2.5
Ea h’s magne ic
lux densi y [6]
B
T
5
5 10
−
⋅
cu en in he cable
I
A
5
5 10
⋅
6
10
6
1.5 10
⋅
* Ex e nal o ce is no conside ed a his s age o esea ch
0
10
20
30
40
50
0 10 20 30 40 50
x (m)
y
(m)
s = 120 m
s = 130 m
s = 140 m
Fig. 6: Dependence o he li
y
o he cable on leng h
s
(
6
1.5 10
I
= ⋅
A)
We can see ha he li inc emen o he cable
dec eases wi h he leng h
s
o he b aked-o cable.
I is ob iously caused by he change o o ien a ion o
ec o
(
)
L
s
along he leng h
s
when his leng h
changes, see Figs. 8 and 9. Inc ease o li
(
)
y x
could be, o cou se, achie ed by diminishmen o
speci ic mass
g
o he cable, by g ow h o cu en
I
and, especially, by g ow h o dis ance PQ
2
l
=
.
3600
3700
3800
3900
4000
4100
4200
4300
4400
0 10 20 30 40 50
x
(m)
F
s
(N)
s = 120 m
s = 130 m
s = 140 m
Fig. 7: Dependence o he in e nal o ce
s
F
on leng h
s
(
6
1.5 10
I
= ⋅
A)
E alua ion o dis ibu ion o he o ces ac ing on
he cable on i s b aked-o leng h
s
and cu en
I
can be ca ied ou om Figs. 7, 8 and 9.
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
x (m)
Lx
(N/m)
s = 120 m
s = 130 m
s = 140 m
Fig. 8: Dependence o componen
L
x
o he Lo en z
o ce
L
on leng h
s
(
6
1.5 10
I
= ⋅
A)
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50
x (m)
L,y
(N/m)
s0 = 120 m
s0 = 130 m
s0 = 140 m
Fig. 9: Dependence o componen
L,
y
o he Lo en z
o ce
L
on leng h
s
(
6
1.5 10
I
= ⋅
A)
Fig. 7 shows ha in e nal o ce
s s
F
= F
p o-
ducing ension in he cable s ongly depends on he
b aked-o leng h
s
o he cable. This is e iden ly
associa ed wi h he balance o o ces ac ing on he
cable. O ien a ion o he ec o o speci ic o ce
L,s
igu ing in he balance changes wi h
s
and
depends on i s alue. Bu i s alue is ( om he iew-
poin o s eng h o Ke la whose Young modulus is
[5]
11
1.24 10
E= ⋅ N/m
2
) qui e accep able.
Figs. 8 and 9 show he quali a i e and quan i a-
i e changes o ec o
L,s
p oduced by il ing o he
cable wi h g ow h o leng h
s
. The ec o is always
pe pendicula o he angen o cu e
(
)
y x
≡s,
which esul s in dec ease o he componen
(
)
L,y
s
,
i.e. he le i a ion componen o he o al Lo en z
o ce
L,y
F
. This ac ep esen s a ce ain es ic ion
o le i a ion e ec s, bu his can be compensa ed –
as said abo e – by p olonging o he o iginal dis-
ance o he cable
2
l
.
Res ic ion o he conside ed le i a ion e ec by
he alue o b aked-o leng h
s
o he cable may be
e alua ed om Figs. 10, 11 and 12.

Ad ances in Elec ical and Elec onic Enginee ing
272
0
10
20
30
40
100 105 110 115 120 125 130 135
s(m)
y
max
=
y
(
x
=0) (m )
0
0,5
1
1,5
2
2,5
F
s,max
= F
s
(x
=0) (kN)
y max
Fs(x=0)
Fig. 10: Dependence o he maximum li
max
y
and in e -
nal o ce
s,max
F
on leng h
s
o he cable (
5
5 10
I
= ⋅
A)
0
10
20
30
40
50
100 105 110 115 120 125 130 135 140 145
s (m)
y
max
=
y
(
x
=0) (m)
0
1
2
3
4
5
F
s,max
= F
s
(x=0) (kN)
y max
Fs(x=0)
Fig. 11: Dependence o he maximum li
max
y
and in e -
nal o ce
s,max
F
on leng h
s
o he cable (
6
10
I=
A)
0
10
20
30
40
50
100 110 120 130 140 150
s (m)
y
max
= y(x=0) (m)
0
1
2
3
4
5
6
7
8
F
s,max
= F
s
(x=0) (kN)
y max
Fs(x=0)
Fig. 12: Dependence o he maximum li
max
y
and
in e nal o ce
s,max
Fon leng h
s
o he cable
(
6
1.5 10
I
= ⋅
A)
The abo e h ee igu es show ha he le i a ion
e ec (in he a angemen o pa ame e s in Tab. 1) is
limi ed. F om ce ain alue o
s
( o example o
5
5 10
I
= ⋅
A om leng h
130
s
≈
m) he o ce
(
)
s
0
x
=
F s ops changing and, he e o e, he bal-
ance o o ces ac ing on he cable emains he same.
Fu he inc ease o i s leng h
s
would lead o in-
c ease o i s weigh , which would b eak he balance
ha is he basic condi ion o success ul le i a ion.
6. CONCLUSION
The pape shows ha he idea o a le i a ing su-
pe conduc i e cable in Ea h’s magne ic ield is,
om he iewpoin o he heo e ical p inciples,
qui e eal. I s p ac ical ealiza ion would equi e,
howe e , solu ion o a numbe o echnological
p oblems (a su icien ly i m and ligh supe conduc-
i e cable, easy gene a ion o high cu en , anspo
o liquid He o he unwound cable e c.) and also
p oblems o economic cha ac e .
Fu he esea ch o heo e ical ques ions in he
domain should be aimed a
• possibili ies o inc ease o he pe pendicula
componen
L
y
F
o he Lo en z o ce by inc ease
o s i ness o a ce ain pa o he cable,
• solu ion o he ask as a ully 3D p oblem ( e-
spec ing o e ec s o wind e c.).
Acknowledgemen
This wo k has been inancially suppo ed om
he G an Agency o he Czech Republic (p ojec
No. 102/04/0095).
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